First, accurate pencil marks
None of what follows works on a grid you have not marked up. Each technique reads patterns in the candidate lists, so a stale or missing candidate does not merely slow you down — it produces a confidently wrong conclusion.
If you have not read the beginner guide, the scanning and singles it covers should be exhausted before you start here. There is no point hunting an X-wing on a grid that still contains three unfound hidden singles.
Sudoku for absolute beginners covers that ground.
Naked pairs, and their larger relatives
If two cells in the same unit hold exactly the same two candidates and nothing else, those two digits are used up by those two cells. You do not know which goes where, and it does not matter: neither digit can appear anywhere else in that unit.
The same argument scales. Three cells in a unit between them holding only three candidates form a naked triple, and the three digits leave the rest of the unit. Four cells with four candidates form a quad. The cells do not each need all the candidates — what matters is that the union of their candidates has the same size as the number of cells.
Its mirror image is the hidden pair: two digits that appear in the candidate lists of only two cells in a unit. Those two cells must take those two digits, so every other candidate can be erased from them. Hidden pairs are harder to see and usually more productive, because they clean out long candidate lists rather than short ones.
The pairs helper
Our board can highlight naked pairs for you from the sidebar. It is there for learning the shape rather than for playing with permanently — the pattern is much easier to recognise unaided once you have seen it flagged twenty times.
Pointing sets and box-line reduction
These two work between a box and a line, and they are the same observation read from either end.
Pointing
If every place a digit could go inside one box sits in the same row, then that digit is definitely somewhere in that row — specifically, inside that box. So it cannot be anywhere else along the row, in either of the other two boxes.
Box-line reduction
Now the same picture from the other direction. If every place a digit could go along one row happens to fall inside a single box, then that box's copy of the digit is on that row — so it can be removed from the rest of the box.
Both are worth checking whenever a digit's candidates look clustered. They rarely solve a cell directly, and they very often unlock the pair or single that does.
The X-wing
This is the first technique that spans the whole grid rather than one unit, and the first that genuinely feels like a discovery when you spot it.
Find a digit that, in two different rows, has exactly two possible positions — and those positions sit in the same two columns in both rows.
The reasoning is a small proof by cases. In the first row the 4 is either in the left column or the right one. If it is on the left, then the second row's 4 must be on the right, because the left column already has one. If it is on the right, the second row's is on the left. Either way both columns get their 4 from these two rows, so no other cell in either column can hold one.
Everything works identically with rows and columns swapped: two columns whose candidate pairs share the same two rows eliminate the digit from the rest of those rows.
They are all one idea
It is worth noticing what these four have in common, because it is easier to remember one principle than four recipes.
In every case you find a set of n cells that must, between them, absorb n specific digits. Once a group of cells has claimed a group of digits, those digits are unavailable to anything else the group can see. A naked pair is two cells claiming two digits inside one unit. A pointing set is a box's candidates claiming a digit for a row. An X-wing is four cells claiming one digit across two rows and two columns.
When you are stuck, the question to ask is not “which technique applies here?” but “is there a group of cells here that has already used up a group of digits?” That question finds patterns nobody has given a name to.
The order to try things in
- Exhaust singles first. Naked and hidden singles are cheap. Never go hunting for something clever while a single is sitting unfound.
- Then pairs, in the unit you last changed. A placement often creates a pair next door.
- Then pointing and box-line, digit by digit. Pick the digit with fewest remaining placements; its candidates are the most clustered.
- Then X-wings, and only for digits with few positions left. Scanning all nine digits for X-wings is slow; scanning the two or three sparse ones is quick.
- If nothing gives, suspect an error. On our hardest setting the grid is still solvable by exactly these methods. A genuine dead end almost always means a wrong digit earlier.
Every grid our dealer produces is verified to have a single solution, so there is always a next deduction. If you cannot find it, turn on conflict highlighting and check your work before assuming you need a technique nobody has taught you.
Go looking for a pair
Set the difficulty to hard, turn notes on, and mark up one row at a time. The first naked pair you find unaided is the point this stops feeling like homework.